Self-dual gravity as a two dimensional theory and conservation laws

TL;DR

This paper reformulates self-dual Einstein equations as a two-dimensional field theory, revealing an infinite set of non-local conserved currents and connecting them to the chiral model with area-preserving diffeomorphisms.

Contribution

It demonstrates how the self-dual Einstein equations can be interpreted as a two-dimensional chiral model with a natural Hamiltonian structure and conservation laws.

Findings

Reformulation of SDE as 2D divergence-free vector fields

Identification of SDE with a 2D chiral model

Derivation of infinite non-local conserved currents

Abstract

Starting from the Ashtekar Hamiltonian variables for general relativity, the self-dual Einstein equations (SDE) may be rewritten as evolution equations for three divergence free vector fields given on a three dimensional surface with a fixed volume element. From this general form of the SDE, it is shown how they may be interpreted as the field equations for a two dimensional field theory. It is further shown that these equations imply an infinite number of non-local conserved currents. A specific way of writing the vector fields allows an identification of the full SDE with those of the two dimensional chiral model, with the gauge group being the group of area preserving diffeomorphisms of a two dimensional surface. This gives a natural Hamiltonian formulation of the SDE in terms of that of the chiral model. The conservation laws using the explicit chiral model form of the equations are also given.